On the maximal distance between triangular embeddings of a complete graph

The distance d(f, f ′) between two triangular embeddings f and f ′ of a complete graph is the minimal number t such that we can replace t faces in f by t new faces to obtain a triangular embedding isomorphic to f ′. We consider the problem of determining the maximum value of d(f, f ′) as f and f ′ range over all triangular embeddings of a complete graph. The following theorem is proved: for every integer s 9, if 4s+1 is prime and 2 is a primitive root modulo (4s+1), then there are nonorientable triangular embeddings f and f ′ of K12s+4 such that d(f, f ′) (1/2)(4s + 1)(12s + 4) −O(s), where (4s + 1)(12s + 4) is the number of faces in a triangular embedding of K12s+4. Some number-theoretical arguments are advanced that there may be an infinite number of odd integers s satisfying the hypothesis of the theorem. © 2005 Elsevier Inc. All rights reserved. MSC: 05C10